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Local Kc constructions

Published online by Cambridge University Press:  12 March 2014

J. R. Steel
J. R. Steel*
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, California 94720, USA. E-mail: [email protected]
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Extract

The full-background-extender Kc -construction of [2] has the property that, if it does not break down and produces a final model , then

Ή is Woodin in V ⇒ Ή is Woodin in ,

for all Ή. It is natural to ask whether

κ is strong in V ⇒ κ is λ-strong in ,

for all κ, or even better,

κ is λ-strong in V ⇒ κ is λ-strong in .

As one might suspect, the more useful answer would be “yes”.

For the Kc-construction of [2], this question is open. The problem is that the construction of [2] is not local: because of the full-background-extender demand, it may produce mice projecting to ρ at stages much greater than ρ. Because of this, there is no reason to believe that if E is a λ-strong extender of V, then The natural proof only gives that if κ is Σ2-strong, then Σ, is strong in .

We do not know how to get started on this question, and suspect that in fact strong cardinals in V may fail to be strong in , if is the output of the construction of [2]. Therefore, we shall look for a modification of the construction of [2]. One might ask for a construction with output such that

(1) iteration trees on can be lifted to iteration trees on V,

(2) ∀δ(δ is Woodin ⇒ δ is Woodin in ), and

(3) (a) ↾κ(κ is a strong cardinal ⇒ κ is strong in ), and (b) ↾κ↾λ(Lim(λ) Λ κ is λ-strong ⇒ κ is λ-strong in ).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 721 - 737
Copyright
Copyright © Association for Symbolic Logic 2007

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References

[1] Andretta, A., Neeman, I., and Steel, J. R., The domestic levels of Kc are iterable, Israel Journal of Mathematics, vol. 125 (2001), pp. 157–201.CrossRefGoogle Scholar
[2] Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees. Lecture Notes in Logic 3, Springer-Verlag, Berlin, 1994.CrossRefGoogle Scholar
[3] Neeman, I., unpublished manuscript.Google Scholar
[4] Neeman, I. and Steel, J. R., A weak Dodd-Jensen lemma, this Journal, vol. 64 (1999), pp. 1285–1294.Google Scholar
[5] Steel, J. R., The core model iterability problem, Lecture Notes in Logic 8, Springer-Verlag, Berlin, 1996.CrossRefGoogle Scholar
[6] Steel, J. R., Derived models associated to mice, Proceedings of the July 2005 Singapore workshop Computational prospects of infinity, to appear.Google Scholar
[7] Steel, J. R., An outline of inner model theory, Handbook of set theory, to appear.Google Scholar